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YOU VE SEEN HOW INDUCTIVE AND CAPACITIVE REACTANCE CAN BE REPREsented along a line perpendicular to resistance. In this chapter, you ll put all three of these quantities R, XL, and XC together, forming a complete, working definition of impedance. You ll also get acquainted with admittance, impedance s evil twin. To express the behavior of alternating-current (ac) circuits, you need two dimensions, because ac has variable frequency along with variable current. One dimension (resistance) will suffice for dc, but not for ac. In this chapter and the two that follow, the presentation is rather mathematical. You can get a grasp of the general nature of the subject matter without learning how to do all of the calculations presented. The mathematics is given for those of you who wish to gain a firm understanding of how ac circuits behave.
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What does the lowercase j actually mean in expressions of impedance such as 4 j7 and 45 j83 This was briefly discussed earlier in this book, but what is this thing, really Mathematicians use the lowercase letter i to represent j. (Mathematicians and physicists/engineers often differ in notation as well as in philosophy.) This imaginary number is the square root of 1. It is the number that, when multiplied by itself, gives 1. So i j, and j j 1. The entire set of imaginary numbers derives from this single unit. The square of an imaginary number is negative always. No real number has this property. Whether a real number is positive, zero, or negative, its square can never be negative never. The notion of j (or i, if you re a mathematician) came about simply because some mathematicians wondered what the square root of 1 would behave like, if there were such a thing. So the mathematicians imagined the existence of this animal, and found that it had certain properties. Eventually, the number i was granted a place among the 264
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Complex numbers 265 realm of numbers. Mathematically, it s as real as the real numbers. But the original term imaginary stuck, so that this number carries with it a mysterious aura. It s not important, in this context, to debate the reality of the abstract, but to reassure you that imaginary numbers are not particularly special, and are not intended or reserved for just a few eccentric geniuses. Imaginary numbers are as real as the real ones. And just as unreal, in that neither kind are concrete; you can hold neither type of number in your hand, nor eat them, nor throw them in a wastebasket. The unit imaginary number j can be multiplied by any real number, getting an infinitude of imaginary numbers forming an imaginary number line (Fig. 15-1). This is a duplicate of the real number line you learned about in school. It must be at a right angle to the real number line when you think of real and imaginary numbers at the same time.
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15-1 The imaginary number line.
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When you add a real number and an imaginary number, such as 4 j7 or 45 j83, you have a complex number. This term doesn t mean complicated; it would better be called composite. But again, the original name stuck, even if it wasn t the best possible thing to call it. Real numbers are one dimensional. They can be depicted on a line. Imaginary numbers are also one dimensional for the same reason. But complex numbers need two dimensions to be completely defined.
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Adding and subtracting complex numbers
Adding complex numbers is just a matter of adding the real parts and the complex parts separately. The sum of 4 j7 and 45 j83 is therefore (4 45) j(7 83) 49 j( 76) 49 j76.
266 Impedance and admittance Subtracting complex numbers works similarly. The difference (4 j7) (45 j83) is found by multiplying the second complex number by 1 and then adding the result, getting (4 j7) ( 1(45 j83) (4 j7) ( 45 j83) 41 j90. The general formula for the sum of two complex numbers (a jb) and (c jd) is (a jb) (c jd) (a c) j(b d)
The plus and minus number signs get tricky when working with sums and differences of complex numbers. Just remember that any difference can be treated as a sum: multiply the second number by 1 and then add. You might want to do some exercises to get yourself acquainted with the way these numbers behave, but in working with engineers, you will not often be called upon to wrestle with complex numbers at the level of nitty-gritty. If you plan to become an engineer, you ll need to practice adding and subtracting complex numbers. But it s not difficult once you get used to it by doing a few sample problems.
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